Skip to main content
×
×
Home

Delineating classes of computational complexity via second order theories with weak set existence principles. I

  • Aleksandar Ignjatović (a1)
Abstract

In this paper we characterize the well-known computational complexity classes of the polynomial time hierarchy as classes of provably recursive functions (with graphs of suitable bounded complexity) of some second order theories with weak comprehension axiom schemas but without any induction schemas (Theorem 6). We also find a natural relationship between our theories and the theories of bounded arithmetic (Lemmas 4 and 5). Our proofs use a technique which enables us to “speed up” induction without increasing the bounded complexity of the induction formulas. This technique is also used to obtain an interpretability result for the theories of bounded arithmetic (Theorem 4).

Copyright
References
Hide All
[1]Buss, Samuel R., Bounded arithmetic, Bibliopolis, Naples, 1986.
[2]Buss, Samuel R. and Ignjatović, Aleksandar, Unprovability of consistency statements in fragments of bounded arithmetic, Technical Report CMU-PHIL-43, Carnegie-Mellon University, Department of Philosophy, Pittsburgh, Pennsylvania, 01 1994; Annals of Pure and Applied Logic (to appear).
[3]Cook, Stephen, Feasibly constructive proofs and the propositional calculus, Proceedings of the Seventh Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 1975, pp. 8397.
[4]Ferreira, Fernando J. I., Polynomial time computable arithmetic, Logic and computation (Sieg, W., editor), Contemporary Mathematics, vol 106, American Mathematical Society, Providence, Rhode Island, 1990, pp. 137156.
[5]Ferreira, Fernando J. I., Polynomial time computable arithmetic and conservative extensions, Ph.D. thesis, Penn-sylvania State University, University Park, Pennsylvania, 1988.
[6]Ferreira, Fernando J. I., Stockmeyer induction, Feasible Mathematics (Buss, S. R. and Scott, P. J., editors), Proceedings of the Mathematical Sciences Institute workshop, Ithaca, New York, 06 1989, Birkhäuser, Boston, Massachusetts, 1990, pp. 161180.
[7]Hájek, P. and Pudlák, P., Metamathematics of first-order arithmetic, Springer-Verlag, Berlin, 1992.
[8]Ignjatović, Aleksandar, Delineating classes of computational complexity via second order theories with weak set existence principles. I, Technical report CMU-PHIL-22, Department of Philosophy, Carnegie-Mellon University, Pittsburgh, Pennsylvania, 11 1991.
[9]Ignjatović, Aleksandar, Induction in theories of bounded arithmetic, manuscript in preparation.
[10]Ignjatović, Aleksandar and Sieg, Wilfried, Herbrand analysis of some theories with weak set existence principles, manuscript in preparation.
[11]Leivant, Daniel, A foundational delineation of computational feasibility, draft (07 1991).
[12]Pudlák, Pavel, A note on bounded arithmetic, Fundamenta Mathematicae, vol. 136 (1990), pp. 8589.
[13]Solovay, Robert, Letter to P. Hájek, August 1976.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 3 *
Loading metrics...

Abstract views

Total abstract views: 59 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 15th August 2018. This data will be updated every 24 hours.